# On general position sets in Cartesian products

## TL;DR

This paper investigates the general position number in Cartesian product graphs, providing exact values for cylinders, bounds for powers, and formulas for grid graphs, advancing understanding of vertex set configurations in these structures.

## Contribution

It introduces new results on the general position number for cylinders, Cartesian powers, and grid graphs, including exact values, bounds, and formulas, expanding the theoretical framework.

## Key findings

- gp(C_r  C_s)   {6,7} for specified r,s
- Probabilistic lower bounds for Cartesian powers
- Explicit formula for gp-sets in P_r  P_s

## Abstract

The general position number ${\rm gp}(G)$ of a connected graph $G$ is the cardinality of a largest set $S$ of vertices such that no three distinct vertices from $S$ lie on a common geodesic; such sets are refereed to as gp-sets of $G$. The general position number of cylinders $P_r\,\square\, C_s$ is deduced. It is proved that ${\rm gp}(C_r\,\square\, C_s)\in \{6,7\}$ whenever $r\ge s \ge 3$, $s\ne 4$, and $r\ge 6$. A probabilistic lower bound on the general position number of Cartesian graph powers is achieved. Along the way a formula for the number of gp-sets in $P_r\,\square\, P_s$, where $r,s\ge 2$, is also determined.